students’ mathematical reasoning: how could it be …

AKSIOMA: Jurnal Program Studi Pendidikan Matematika ISSN 2089-8703 (Print)

Volume 10, No. 3, 2021, 1477-1492 ISSN 2442-5419 (Online)

DOI: https://doi.org/10.24127/ajpm.v10i3.3783

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STUDENTS’ MATHEMATICAL REASONING: HOW COULD IT BE

THROUGH MHM-PROBLEM BASED STRATEGY AIDED INTERACTIVE

MULTIMEDIA?

Runisah1*

, Wiwit Damayanti Lestari2, Nurfadilah Siregar

3

1,2

Universitas Wiralodra, Indramayu, Indonesia 3 Universitas Tanjungpura, Pontianak, Indonesia

*Corresponding author. Jl. Ir. H. Juanda, Km 3, 45213, Indramayu, Indonesia E-mail: [email protected] 1*)

[email protected] 2)

[email protected] 3)

Received 04 June 2021; Received in revised form 13 September 2021; Accepted 28 September 2021

Abstrak

Penelitian ini bertujuan untuk mengetahui kemampuan penalaran matematis siswa yang memperoleh

pembelajaran Mathematical Habits of Mind-Problem Based Strategy (MHM-PB) berbantuan multimedia

interaktif, untuk mengetahui pengaruh MHM-PB terhadap kemampuan penalaran matematis siswa

berdasarkan jenis kelamin, dan untuk menganalisis kesulitan siswa dalam menyelesaikan tes kemampuan

penalaran matematis. Penelitian ini dilakukan dengan menggunakan metode eksperimen semu dengan

desain kelompok kontrol pre-test dan post-test. Data diperoleh dari 66 siswa kelas VII di Kabupaten

Indramayu, Indonesia, dengan menggunakan lima soal tes uraian terkait kemampuan penalaran

matematis. Strategi MHM-PB berbantuan Multimedia Interaktif digunakan di kelas eksperimen dan kelas

lainnya memperoleh pembelajaran konvensional. Hasil penelitian menunjukkan bahwa penalaran

matematis siswa yang menggunakan strategi MHM-PB berbantuan multimedia interaktif lebih baik

dibandingkan dengan strategi konvensional. Tidak terdapat perbedaan kemampuan penalaran matematis

berdasarkan jenis kelamin pada siswa yang menggunakan strategi MHM-PB. Selain itu, beberapa siswa

masih mengalami kesulitan dalam membuat kesimpulan, memberikan alasan atau bukti atas jawaban yang

mereka berikan, dan memeriksa kebenaran suatu pernyataan. Sementara itu, membuat generalisasi

merupakan kesulitan yang banyak dialami siswa.

Kata kunci: Habits of mind; multimedia interaktif; penalaran matematis.

Abstract

This study aims to determine students' mathematical reasoning ability using Mathematical Habits of

Mind-Problem based Strategy (MHM-PB) strategy aided interactive multimedia, to analyze the effect of

using MHM-PB on mathematical reasoning abilities based on gender differences, and to analyze

students' difficulties in solving mathematical reasoning ability tests. This research was carried out using

the quasi-experimental method with pre-test and post-test control group design. Data were obtained from

66 grade VII students at Indramayu Regency, Indonesia using an essay test with five problems on

mathematical reasoning ability. Mathematical Habits of Mind-Problem Based Strategy aided Interactive

Multimedia is used in experimental group and the other group received conventional strategy. The result

showed that students’ mathematical reasoning using MHM-PB strategy aided interactive multimedia was

better than the conventional strategy. There is no difference in mathematical reasoning abilities based on

gender in students who use MHM-PB. Furthermore, some students still have difficulty making a

conclusion, providing reasons or evidence for the answers they give, and checking the truth of a

statement. Meanwhile, making generalizations is a difficulty that many students experience.

Keywords: Habits of mind; Interactive multimedia; Mathematical Reasoning

This is an open access article under the Creative Commons Attribution 4.0 International License

mailto:[email protected]



http://creativecommons.org/licenses/by/4.0/




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INTRODUCTION

Reasoning is a process of thinking

to draw logical conclusions from facts,

information in various ways that truth is

recognized. According to Tanisli (2016)

and Rizqi & Surya (2017), reasoning is

a thinking process used to make

conclusions or make a new statement based on prior information. Meanwhile,

Yanto et al. (2019) stated that the

process of reasoning includes linking

evidence and facts to construct logical

conclusions. Rohana (2015) specifically

stated that mathematical reasoning is

used to draw conclusions and solve

mathematical problems based on logical

and critical facts.

Reasoning also enables students

to determine various ideas from facts or

use various existing information to

solve mathematical problems.

According to the National Council of

Mathematics Teachers (NCTM, 2000),

mathematical activities are inseparable

from reasoning because it plays a vital

role in solving problems (Rohana, 2015;

Napitupulu et al., 2016; Hasanah et al.,

2019). Mueller & Maher (2009) stated

that reasoning forms the basis of

mathematical understanding. Therefore,

it is needed by students in

understanding, solving, and learning

various mathematics concepts.

The importance of reasoning in

mathematics learning activities is one of

the objectives of teaching mathematics

to students. To teach students the

reasoning is one of the important goals

in mathematics Jeannotte & Kieran

(2017). After students learn the subjects

at the primary and secondary education

level, they are expected to possess

mathematical reasoning, such as making

generalizations, guesses and verifying

them based on patterns, facts,

phenomena, or existing data

(Kemdikbud, 2017).

According to Isnaeni et al. (2018),

students’ mathematical reasoning ability

is still low irrespective of the

importance of possessing such a skill.

This is in addition to the numerous

studies on mathematical reasoning,

which indicated low mathematical

reasoning ability. The yearly results of

the Program for International Students

Assessment (PISA) test for the

mathematics category from year to year,

Indonesia's achievements are still lower

than other participating countries

(OECD, 2019; Nizam, 2016; Pratiwi,

2019). Furthermore, the Trends in

International Mathematics and Science

Study (TIMSS) study results from 1999

to 2015 (Nizam, 2016; Mullis et al.,

2016) showed the same. Furthermore,

several research results indicated

differences in abilities between male

and female students in their reasoning

abilities. In language, female students

are superior to males, but male students

are superior in science and reasoning

(Kuhn & Holling, 2009). Gender affects

students’ understanding of mathematics

(Fajar, 2016). Male students have

superior reasoning ability than female

students (Setiawan & Sajidah, 2020).

Several other studies have shown

various problems related to reasoning.

The results showed that teachers had

difficulty in generalizations (Moguel et

al., 2019). Students can make mistakes

in solving the problems analogies (K.

Saleh et al., 2017). Students can make

mistakes in every stage of reasoning. It

performs mathematical manipulations

and provides a reason or evidence to the

truth of the solution, checking the

validity of an argument and conclusion (Setiawan & Sajidah, 2020).

Habits are used to determine

students' mathematical reasoning abili-

ty. According to Mahmudi & Sumarmo

(2015), positive habits carried out by




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students consistently have the potential

to form a variety of positive abilities.

Furthermore, one of the strategies that

emphasize students’ thinking habits is

the Mathematical Habits of Mind-

Problem Based (MHM-PB). Learning

with MHM-PB is integrating problem-

based learning with the Mathematical

Habits of Mind (MHM) strategy.

Jacobbe & Millman (2009)

carried out research to determine

students thinking habit in mathematics

to 1) explore ideas, 2) formulate

questions, 3) construct examples, 4)

identify problem-solving approaches

that are useful in large classes, 5)

inquire about the possibility of

“something more” (a generalization) in

the content on which they are working,

and 6) reflect on their answer to

determine the possibility of errors

known as MHM (Miliyawaty, 2014).

Thus, the MHM strategy has the

potential to develop students’ thinking

abilities maximally.

The Problem-Based Learning

model has a procedure consisting of the

following: 1) the teacher presents the

problem to the students, 2) the students

identify the given problem, 3) they seek

information from various sources, 4)

they choose the most appropriate

solution, and 5) the teacher evaluates

the students' work (Gorghiu et al.,

2015). By paying attention to the

procedures in Problem-Based Learning,

the model promotes students to use their

reasoning in solving problems.

Although studies are rarely

conducted on the MHM-PB strategy,

previous research indicates that

students’ creative thinking abilities can

be improved through this process

(Andriani et al., 2017; Mahmudi &

Sumarmo, 2015). Furthermore,

according to Mahmudi & Sumarmo

(2015), students taught with the MHM

strategy perform better in terms of

solving mathematical problems. In line

with other studies show the impact of

implementing MHM on children with

challenging behaviors, such as

increased task persistence, application

of knowledge in facing new situations,

listening to others with understanding

and empathy, increased managing

impulsivity, and thinking flexibly

(Burgess, 2012).

Another factor supporting the

implementation of teaching and learning

is media, such as interactive

multimedia. According to Khoiri et al.

(2013), multimedia is a tool capable of

creating dynamic and interactive pre-

sentations that combine text, graphics,

animation, audio, and images. Thus,

this research aims to examine:

1) Mathematical reasoning ability of

students using MHM-PB strategy

aided interactive multimedia.

2) The effect of using MHM-PB on

mathematical reasoning abilities is

based on gender differences.

3) Student’s difficulties in answering

tests related to mathematical

reasoning ability.

METHOD

The method used in this research

was quantitative with a quasi-

experimental design. The random cluster sampling was used to obtain data

from 66 grade VII students of Junior

High School in Indramayu, West Java,

Indonesia. The students were grouped

into two equal classes with the same

number of students, with one taught

using MHM-PB strategy aided

interactive multimedia and the other

used conventional learning strategy.

Meanwhile, if viewed from

gender, the subjects are 66 students

consisting of 39 female and 27 male

students. In the experimental group,




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there were 11 male students and 22

female students. In the control group,

there were 16 male students and 17

female students.

In this study, MHM-PB strategy

steps are:

1) The teacher explains the purpose of

the following learning problem

through a PowerPoint slide show,

and students are directed to ask

questions related to the problem.

2) Students gather information to solve

problems in groups by defining and

organizing learning tasks related to

the problems.

3) The teacher encourages students to

discuss in groups, conduct

experiments, explore mathematical

ideas, construct examples, and

formulate hypotheses.

4) Students work on the report of solved

problems by matching the answer to

the solution on the slide.

5) The teacher helps students review the

problem-solving results and evaluate

the process by asking them to present

their work.

6) Through, discussion the teacher and

students identify problem-solving

strategies that can be applied to other

problems.

7) The teacher and students conclude

about the studied concept or material.

The instrument used was a test of

mathematical reasoning ability, which

consists of 5 essay questions and

indicators as follows: 1) drawing con-

clusions, compiling evidence, providing

reasons for the correctness of the soluti-

on, 2) Checking the truth of statement

3) Posing conjecture, 4) Finding patterns or properties of mathematical

symptoms to make generalizations

These indicators were based on trials’

results valid and reliable tests with a

reliability coefficient of 0,57.

To determine students' mathemati-

cal reasoning abilities, the results of the

reasoning ability tests were used after

the entire learning process ended.

Furthermore, the formulation from

Meltzer (2002) was used to determine

the increase of mathematical reasoning

ability. Meanwhile, Hake (1999) was

classified gain is used to interpret

Normalized Gain (N-gain). The

normalized gain is obtained from the

comparison between the difference

between the pretest score and the

posttest score with the difference

between the ideal score and the pretest

score, which can be written as follows.

(1)

With interpretation: (a) high, if

; (b) moderate, if

; (c) low, if .

Furthermore, quantitative data

were analyzed through inferential

statistical analysis. In the inferential

statistical analysis stage, several tests

were used that correspond to the

characteristics of the data (normally

distributed, homogeneous). This stage is

carried out to test the hypothesis

proposed in the study. Prerequisite test

of parametric statistics on mathematical

reasoning abilities of students. The data

are grouped based on learning and

gender. The hypothesis tests used

include two-way ANOVA test and

continued with Sceffe test’. Meanwhile,

to analyze students’ difficulties in

solving problems related to

mathematical reasoning can be seen

from students’ answers.

RESULTS AND DISCUSSION

Results of pre-test, post-test, and

N-gain are shown in Table 1 and Table

2.




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Table 1. Results of pre-test, post-test,

and N-gain in experiment group Experiment Group

Pre-test Post-test N-gain

Maximum score 11 19 0,94

Minimum Score 2 14 0,54

Mean 5,76 17,36 0,81

Standard Deviation 2,60 1,32 0,10

Table 2. Results of pre-test, post-test,

and N-gain in control group Experiment Group

Pre-test Post-test N-gain

Maximum score 11 17 0.78

Minimum Score 2 11 0,25

Mean 5,73 13,88 0,57

Standard Deviation 2,23 1,82 0,14

Table 1 and Table 2 showed that

the experiment group and control group

post-test’s result had difference mean of

3,48. This means that students’ average

mathematical reasoning ability in the

experimental group is higher than

control group, while the ideal maximum

score is 20. Furthermore, based on the

post-test results compared to the ideal

maximum score, the average score for

the experimental group is 86,8%, and

the control group is 69,4%. This acquisition supports the differences in

the increase in mathematical reasoning

abilities between the two groups. The

mean N-gain of the experiment group

means a high increase. Meanwhile, the

mean of N-gain of control group means

on the moderate level based on

(Meltzer, 2002) research.

Data processing was performed to

test the normality of the N-gain data

distribution using the Windows program

SPSS. Therefore, based on Shapiro

Wilk’s test, it can be concluded that the

normality of distribution is fulfilled, or

the population is normally distributed.

Levene’s test indicates that the variance

data is homogeneous. Thus, from two-

way ANOVA test, it can be concluded

that the learning model has a significant

effect on the increase of mathematical

reasoning ability. This is indicated by

the value of F = 62, 95 with the

probability (sig.) is 0,000, that is

smaller than 0,05. This is supported by

the results of the two-way ANOVA test

on the final test results for mathematical

reasoning abilities obtained F = 74,69

with the probability (sig.) is 0,000 that

is smaller than 0,05, which shows the

existence of different reasoning abilities

between the experimental and control

groups. In this case, the mathematical

reasoning abilities of students who use

MHM-PB are better than students who

use conventional models. This means

that the MHM-PB learning model

affects students’ mathematical

reasoning abilities.

Furthermore, regardless of the

learning model used, the final test

results are obtained F = 0,106 with the

probability (sig.) is 0,746, greater than

0,05. This means that male and female

students have the same reasoning

abilities. These results support the

results of the test results in which

increased reasoning abilities have

obtained the value of F = 0,27 with the

probability (sig.) is 0,61 that is greater

than 0,05. This means that there is no

difference in the increase in

mathematical reasoning abilities

between male and female students.

Based on the results of further tests with

the Scheffe test, the sig. value was

obtained 0,976 greater than 0,05, so

there is no difference in reasoning

ability between male students and

female students in the group of MHM-

PB strategy.

The results show that the

mathematical reasoning abilities of

students who get MHM-PB are better

than students who use the conventional

model. This happens because of the




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various advantages of the MHM-PB

model. Through the MHM-PB strategy,

students are accustomed to constructing

or making examples, exploring

mathematical ideas, making

generalizations, and solving

mathematical problems. This is

confirmed in NCTM (2000) that

mathematical reasoning occurs when

the learner: 1) observe a pattern, 2)

formulate generalization and conjecture

related to observed regularity, 3)

assess/test the conjecture; 4) construct

and assess mathematical arguments, and

5) describe (validate) logical

conclusions about some ideas and its

relatedness. This is also in line with the

opinion of experts that reasoning works

when someone tries to understand

problems, make relationships and

representations between concepts, as

well as assumptions and generaliza-

tions, to prove these allegations

(Napitupulu et al., 2016; Hasanah et al.,

2019). Students’ reasoning abilities are

built when they are involved in the

problem-solving process. Positive

habits that are consistently carried out

can develop positive abilities, with

thinking habituation capable of spurring

students to build reasoning ability

(Mahmudi & Sumarmo, 2015).

Constructing examples as part of

learning with MHM-PB has many

benefits in improving students'

reasoning abilities. According to

Dreyfus et al. (2006) constructing

examples is a complex task that requires

students to make connections between

concepts. Students may make incorrect

generalizations if students are not

allowed to construct examples and non-examples (Miliyawati, 2014). Making

proper generalizations through MHM-

PB indicates the students' good

reasoning ability when allowed to make

examples.

The habits of exploring

mathematical ideas in learning with the

MHM -PB strategy enable students to

determine the relationship between

various mathematical concepts.

According to Miliyawati (2014), the

MHM strategy promotes students to

make connections between

mathematical ideas, which is one of the

advantages of MHM-PB compared to

conventional learning.

Students’ ability to collaborate to

conduct exploration and challenges

during the MHM-PB strategy promotes

meaningful learning. The research

obtained several attributes that promote

meaningful mathematics learning,

specifically to ensure: a) students are

challenged and active, b) the teacher

pays attention to the development of

students' ideas, c) appropriate and open

tasks, d) collaboration and e) there are

good appreciation and acceptance of

ideas, conjectures, and other alternatives

given by students (Mueller et al., 2014).

According to Mahmudi & Sumarmo

(2015), student learning activities with

problem-based MHM strategies provide

opportunities for developing their actual

and potential abilities following

Vygotsky's theory. Furthermore, Yackel

& Cobb (1996) stated that a learning

community is formed where students

learn actively, provide, respond and

defend emerging ideas in a discussion.

Mathematical reasoning and

understanding naturally arise from

communication in such communities.

In the MHM-PB strategy, the

teacher acts as a facilitator to guide

students during group discussions.

When students do not understand a topic, the teacher does not give direct

answers. Instead, they provide probing

or guiding questions, such as asking

them to explain their thinking, offer

evidence, and use previous knowledge




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to deal with problems that arise.

According to Mueller et al. (2014),

probing questions provide a deeper

conceptual understanding and enables

students to connect previous knowledge

with new ideas. Through guiding

questions, the teacher tries to guide

students in solving problems by asking

for solutions, procedures, or strategies.

Furthermore, it strengthens students’

conceptual understanding and supports

them in creating their heuristics

(Mueller et al., 2014). Meanwhile,

interactive media in learning with the

MHM-PB strategy has various

advantages, such as facilitating students'

understanding (Nickchen & Mertsching,

2016). This is due to the strong

relationship between students’ under-

standing and reasoning (Bakar et al.,

2018). Similarly, Hwang et al. (2015)

stated that interactive media could

develop students' mathematical abilities.

In conventional learning, the

teacher provides concepts or materials

directly to students and then draws

questions with the solution, followed by

exercises. In this strategy, they learn by

paying attention to the teacher during

the learning activities. Furthermore,

they are not allowed to participate

actively. Therefore, the learning

atmosphere feels boring, and various

cognitive aspects possessed by students

are less developed, including

mathematical reasoning.

The result showed that students

that use the MHM-PB strategy are more

active in exploring and solving

problems presented on worksheets.

Meanwhile, those with conventional

learning are less involved in thinking

activities to explore new ideas related to

the studied concepts. The results of this

research are in line with previous

studies. For instance, Dwirahayu et al.

(2018) stated that Habits of Mind

positively influence mathematical

ability generalization. MHM strategy

allows students to think logically,

systematically, accurately, and critically

(Hafni et al., 2019). This research is

also in line with the previous studies

carried out by Napitupulu et al. (2016),

Siregar et al. (2017), Bernard &

Chotimah (2018), Saleh et al. (2018),

which uses constructivism-based

learning to improve students’ mathema-

tical reasoning ability. The study

successively uses Problem-based

learning, MCREST strategy, an open-

ended approach using VBA for Power

Point, and RME.

Furthermore, without paying

attention to the learning model, the

results of this study indicate that there is

no difference in mathematical reasoning

abilities between male and female

students. The results of this study are in

line with the results of the study Salam

& Salim (2020), which states that if you

ignore the learning model, used

mathematical reasoning abilities

between male and female students do

not differ significantly. Furthermore, the

students who used the MHM PB

strategy of male and female students'

mathematical reasoning abilities did not

differ significantly. This is in line with

(Kadarisma et al., 2019) stated that is no

significant difference in mathematical

reasoning abilities between male and

female students after using a problem-

based learning approach. Thus, the

MHM-PB Strategy can minimize

differences in mathematical reasoning

between male and female students.

In MHM-PB, discussions carried

out to explore mathematical ideas or

solve problems are carried out in small

groups consisting of students with

different abilities and genders. This can

reduce the ability of male and female

students to reason. The division of




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small heterogeneous learning groups is

one factor that causes no difference in

mathematical reasoning ability between

men and women (Kadarisma et al.,

2019). Small groups from diverse

backgrounds can help overcome social

barriers among students and allow

collaborative learning among them

(Argaw et al., 2017). In order to be

active in group discussion and exercise

independent learning, students need to

develop social skills (Ulger, 2018).

Furthermore, the study

determined several weaknesses

possessed by most students, as indicated

in their answers can be seen in the

following description.

Problem 1 The properties of a triangle are known

as follows:

a. Has 2 equal sides.

b. Has 2 angles of the same size.

c. Has 1 axis of symmetry and 1

rotational symmetry.

d. Occupy its frame in 2 ways.

From the above statement, we can

conclude what the triangle is?

Examples of student answers can be

seen in Figure 1.a and Figure 1.b.

Figure 1.a. Examples of students’

wrong answer

Figure 1.b. Examples of students’

correct answer

In Figure 1.a, the student did not

answer. He only wrote back the

properties of the triangle that were

written in the question. This shows that

students have not understood the

triangle concept well, so they are weak

in reasoning and checking the truth.

To make it easier to solve these

problems, one way to sketch an image

based on the information provided.

Making a written presentation of ideas

into pictures will help students organize

their thoughts, but they do not do it.

This indicates that students have

weaknesses in representing written

ideas in the form of images that will

help them answer Problem 1.

According to Noto et al. (2016), the

right of representation makes

mathematical ideas more concrete, and

complex problems become simpler so

that they are easier to solve. Meanwhile,

in Figure 1.b, the students concluded

that the triangle that fulfills the

characteristics described in the problem

is isosceles. To make it easier to make

conclusions, students sketch images

from the data provided in the questions.

Problem 2 Are all equilateral triangles right

triangles? Explain!

Example of student answer for Problem

2 can be seen in Figure 2.

Figure 2. Examples of students’ wrong

answers

In the Figure 2, it appears that

students are giving reasons for wrong

answers. The student explains that a




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right triangle is a triangle whose sides

are the same length, and one angle is a

90o right angle. From these answers,

these students do not understand the

concept of a right triangle. A triangle

with both sides is the same length, and

one of the angles measuring 90o as

explained by the student, is an isosceles

right triangle. Students do not

understand that a right triangle should

not have two sides of the same length.

This means that students do not

understand the types of triangles

(equilateral, isosceles, and right

triangle) as a whole and the relationship

between these triangles. The lack of

understanding of these students causes

student errors in providing reasons for

the answers given by students to the

problems. According to Strand et al.

(2006) lack of understanding of the

basic concepts of a topic fails to use

formal procedures to solve several types

of problems and it differences based on

gender.

Another example of students’

answer in Problem 2 can be seen in

Figure 3.

Figure 3. Examples of student answers

Students give correct answers, but

the reasons given are not clear.

According to the students, an equilateral

triangle has an angle of 60o, and a right

triangle is 90o. The answer is not clear

whether all the angles are 60o or if one

of the corners has a magnitude of 60o. If

the triangle is only one of the corners

that has a large 60o, then it is still

possible that the other angles have a

large 90o and 30

o. Such a triangle is a

right triangle. Thus, it appears that

students are less able to communicate

their ideas in writing. This is in line

with research Sumaji et al. (2019),

students have problems communicating

given problems, and students have

problems communicating mathematical

problems in the form of the written text

Problem 3 Given a rectangle.

DC length 8 cm and CB length 6 cm,

then:

a. BD length is 10 cm. Is that right?

Prove it!

b. The area of triangle BCD is 24 cm2.

Is that right? Prove it.

From the results of the students’

answers, some students answered

incorrectly. Figure 4 is an example of a

student's wrong answer.

Figure 4. Examples of students' wrong

answers

The student answered incorrectly

to question Problem 3.b. The student

determined the area by adding up the

length of the sides of the triangle BCD.

In other words, the student looks for the

area using the concept of the perimeter

of the triangle. From the calculation

results obtained 24 cm, these results are

considered by students as the area of the

triangle. From this answer, it can be

A B

DC




1486|

seen that students do not understand the

rules for area and the rules for the

perimeter of a triangle and the concept

of units of length and units of area. The

student does not understand that the 24

cm he gets from the calculation is the

perimeter of the triangle, while the area

of the triangle is (6x8) / 2 = 24 cm2. The

student's lack of understanding of these

basic concepts causes student errors in

solving problems. This is in line with

(Strand et al., 2006). As previously

explained, students' failure to solve

problems is caused by a lack of

understanding of the basic concepts

Problem 4 Given 4 sets of logs with the following

length.

Can every set of logs form a triangle?

Give reasons!

One of the examples of students’

answers given by the majority is shown

in Figure 5.

Figure 5. Example students’ answer

Figure 5 shows that students did

not give correct answers because they

stated that triangles could not be formed

with side lengths of 3, 5, and 7 units

(the length of one log represents, in this

case, one unit). The conclusion is only

based on checks made using the

Pythagorean rule, which only applies to

a right triangle. When a check is carried

out using the triangle’s properties, the

length of wood 3, 5, and 7 units can be

arranged. This is because the two sides’

length is more than the other side and

similar to the sets of logs whose lengths

are 3, 3, and 7. In this case, students

provided answers with the wrong

reasons by using Pythagoras’ rules,

which did not link to the triangle’s other

properties.

Students’ errors in solving

Problem 4 show their weakness

associated with a mathematical

understanding of using the triangle rule

and the Pythagorean formula. In other

words, students' reasoning abilities are

supported by mathematical

understanding. This study’s results align

with (Bakar et al., 2018), research on

the strong positive relationship of

mathematical concept understanding

and reasoning. Napitupulu et al. (2016)

stated that students have difficulties

constructing proof due to a lack of

understanding of the materials that need

to be applied. Most students with low

reasoning abilities have weaknesses in

providing examples in solving

problems, compiling evidence, checking

the validity of answers, and drawing

conclusions (Hasanah et al., 2019).

Problem 5 Given several matchsticks are used to

form equilateral triangles as in

following table.

The number of

matchsticks 3 5 7 9 .…

The number of

Triangles

…

……

…

…...




| 1487

Find the pattern of the relationship

between the number of matchsticks and

the number of equilateral triangles that

can be formed!

Of the answers provided by

students, only a few were correct and

under the relationship pattern. Figure 6

shows an example of a student’s wrong

answer.

Figure 6. Examples of students’ answers

Figure 6 shows that students

already see the formation of several

matches with triangles in the first

pattern and failed to write other patterns

of relationships. In this case, students

still have difficulty determining the

relationship pattern between triangles

and matchsticks. This means that they

are still finding it difficult to generalize.

Napitupulu et al. (2016) stated that the

mistakes and difficulties of students in

answering the mathematical reasoning

ability test are due to 1)

misinterpretation or drawing illogical

conclusions despite being aware of the

assignment demands, 2) lack of

metacognitive processes, 3) Unable to

build meaningful relationships between

the available facts and the objectives, 4)

inability to build data-based or pattern-

based conjectures, and 5)

misconceptions of deductive and

inductive thinking. Moguel et al. (2019)

showed that mathematics teachers had

difficulty observing regularity,

determining the patterns, and

formulating generalizations.

Based on the research results, the

MHM-PB strategy assisted by

interactive multimedia, which is applied

in mathematics learning, has a

tremendous impact in helping develop

students’ mathematical reasoning

abilities. These results can be a

reference for teachers in developing

mathematical reasoning skills, namely

by applying the MHM-PB strategy

assisted by interactive multimedia in

learning mathematics. In addition, the

findings of student errors in working on

the reasoning ability questions on this

triangle material indicate that the

common understanding of concepts

affects students’ reasoning so that it can

be used as material for evaluation and

reflection, especially for junior high

school mathematics teachers to

emphasize understanding concepts in

the learning carried out. The limitation

of this research is the absence of

computer equipment to support

interactive multimedia in research. The

interactive multimedia used based on

Microsoft PowerPoint is only operated

by researchers, not by students.

CONCLUSION AND SUGGESTION In conclusion, students taught

with mathematical habits of mind-

problem-based strategy aided

interactive multimedia learning to have

better mathematical reasoning abilities

than those with conventional

learning. The mathematical reasoning

abilities of male and female students

using the MHM-PB strategy were not

significantly different. Furthermore,

some students still had difficulty

making conclusions, providing reasons

or answers to the questions they gave,

and checking the correctness of

statements. The difficulty that many




1488|

students do is that it is difficult to see

the regularities in the data given to

determine patterns and formulations.

The difficulty of these difficulties is

caused by the weakness of students to

represent written ideas in the form of

images to help solve problems, the

weak ability of students to understand

basic concepts, the weak ability to

determine linkages between concepts,

the weak ability of students to

communicate an idea.

The MHM-PB aid interactive

media strategy can be used as a learning

strategy to improve students’

mathematical reasoning abilities due to

its various advantages. In addition,

MHM-PB can minimize differences in

the reasoning abilities of male and

female students. Therefore, research on

the impact of using MHM PB aid

interactive media on various

mathematical thinking abilities needs to

be conducted. Furthermore, it is

necessary to research ways to overcome

student difficulties in tests of

mathematical reasoning abilities. In

addition, due to the growing

development of various IT-based

learning media, further research is

needed on the effectiveness of using

interactive media as a tool for the

MHM-PB strategy.

ACKNOWLEDGMENT The authors are grateful to

Universitas Wiralodra that supported

this research and SMP N 1 Sindang

mathematics teachers for their

assistance during this research process.

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